A254315 -id:A254315 - cp's OEIS frontend

A254318 Hyper equidigital numbers.

2, 3, 4, 5, 7, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 28, 29, 31, 32, 35, 36, 37, 39, 41, 43, 46, 47, 49, 50, 53, 54, 58, 59, 61, 64, 67, 69, 71, 72, 73, 79, 81, 83, 89, 92, 93, 97, 98, 100, 101, 103, 104, 105, 106, 107, 109, 113, 116, 119

Offset: 1

Michel Lagneau, Jan 28 2015

The distinction between the equidigital numbers (A046758) is that only the distinct digits are counted instead of all digits. Hence the definition:

Write n as product of primes raised to powers, let D(n) = total number of distinct digits in product representation (number of distinct digits in all the primes and number of distinct digits in all the exponents that are greater than 1) and nbd(n) = A043537(n) = number of distinct digits in n; sequence gives n such that D(n) = nbd(n).

			116 is in the sequence because 116 = 2^2*29 => D(116)= A043537(116)=2.

Michel Lagneau, Table of n, a(n) for n = 1..10000

Cf. A043537, A046760, A046758, A046759, A254315, A254317, A254319, A254321.

Cases[Range[400], n_ /; Length[Union[Flatten[IntegerDigits[FactorInteger[n] /. 1 -> Sequence[]]]]]==Length[Union[Flatten[IntegerDigits[n]]]]]

for(n=1,100,s=[];F=factor(n);for(i=1,#F[,1],s=concat(s,digits(F[i,1]));if(F[i,2]>1,s=concat(s,digits(F[i,2]))));if(#vecsort(digits(n),,8)==#vecsort(s,,8),print1(n,", "))) \\ Derek Orr, Jan 30 2015

A254317 a(n) is the least number k such that the number of distinct digits in the prime factorization of k is n (counting terms of the form p^1 as p).

Original entry on oeis.org

1, 6, 26, 102, 510, 3210, 22890, 153690, 1507290, 15618090

Offset: 1

Michel Lagneau, Jan 28 2015

Write k as product of primes raised to powers; then a(n) is the least number k such that the total number of distinct digits in the product representation of k (number of distinct digits in all the primes and number of distinct digits in all the exponents that are greater than 1) is equal to n. The first term a(1)= 1 by convention. The sequence is complete.

Property: all exponents are equal to 1 (see the examples below).

			a(1)  =        1;
a(2)  =        6 = 2*3            and A254315(6)        =  2;
a(3)  =       26 = 2*13           and A254315(26)       =  3;
a(4)  =      102 = 2*3*17         and A254315(102)      =  4;
a(5)  =      510 = 2*3*5*17       and A254315(510)      =  5;
a(6)  =     3210 = 2*3*5*107      and A254315(3210)     =  6;
a(7)  =    22890 = 2*3*5*7*109    and A254315(22890)    =  7;
a(8)  =   153690 = 2*3*5*47*109   and A254315(153690)   =  8;
a(9)  =  1507290 = 2*3*5*47*1069  and A254315(1507290)  =  9;
a(10) = 15618090 = 2*3*5*487*1069 and A254315(15618090) = 10.

Cf. A043537, A254315.

with(ListTools):
for n from 2 to 10 do:
  ii:=0:
  for k from 2 to 10^9 while(ii=0)do:
    n0:=length(k):lst:={}:x0:=ifactors(k):
    y:=Flatten(x0[2]):z:=convert(y,set):
    z1:=z minus {1}:nn0:=nops(z1):
     for m from 1 to nn0 do :
      t1:=convert(z1[m],base,10):z2:=convert(t1,set):
      lst:=lst union z2:
     od:
     nn1:=nops(lst):
     if nn1=n then ii:=1:printf ( "%d %d \n",n,k):
     else
     fi:
  od :
od:

f[n_] := Block[{pf = FactorInteger@ n, i}, Length@ DeleteDuplicates@ Flatten@ IntegerDigits@ Rest@ Flatten@ Reap@ Do[If[Last[pf[[i]]] == 1, Sow@ First@ pf[[i]], Sow@ FromDigits@ Flatten[IntegerDigits /@ pf[[i]]]], {i, Length@ pf}]]; b = -1; Flatten@ Last@ Reap@ Do[a = f[n]; If[a > b, Sow[n]; b = a], {n, 10^6}] (* Michael De Vlieger, Jan 29 2015 *)
With[{s = Array[CountDistinct@ Flatten@ IntegerDigits[FactorInteger[#] /. {p_, e_} /; e == 1 :> {p}] &, 10^6]}, Map[FirstPosition[s, #][[1]] &, Range@ Max@ s]] (* Michael De Vlieger, Nov 03 2017 *)

a(n)=for(k=1,10^5,s=[];F=factor(k);for(i=1,#F[,1],s=concat(s,digits(F[i,1]));if(F[i,2]>1,s=concat(s,digits(F[i,2]))));if(#vecsort(s,,8)==n,return(k)))
print1(1,", ");for(n=2,7,print1(a(n),", ")) \\ Derek Orr, Jan 30 2015

a(10) corrected by Giovanni Resta, Nov 03 2017

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A254318 Hyper equidigital numbers.

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